Finite Group Databasebeta

← #10877312

Subgroups — #10877312

Order 768.

788 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2324C211yesmaximal, characteristic
2#82233C211yesmaximal, characteristic, complemented
3#71678C211yesmaximal, characteristic
4#71677C211yesmaximal, characteristic
5#71605C211yesmaximal, characteristic, complemented
6#71604C211yesmaximal, characteristic, complemented
7#71583C211yesmaximal, characteristic, complemented
8#69656C211yesmaximal, characteristic, complemented
9#69654C211yesmaximal, characteristic, complemented
10#69613C211yesmaximal, characteristic, complemented
11#69611C211yesmaximal, characteristic, complemented
12#69506C211yesmaximal, characteristic, complemented
13#69505C211yesmaximal, characteristic, complemented
14#69468C211yesmaximal, characteristic, complemented
15#69467C211yesmaximal, characteristic, complemented
16#27688—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C3×#390—21no—
23C3×#390C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#388—21no—
26C3×#386—21no—
27C3×#385—21no—
28C3×#380C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#4984—21no—
37#4984—21no—
38C2×#808C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4950—21no—
43#4950—21no—
44#4936C2211yescharacteristic
45#4936C2211yescharacteristic
46#4935C2211yescharacteristic
47#4935C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2324D311yescharacteristic
61#2317—31no—
62#2101—31no—
63#2099—31no—
64#2094—31no—
65#2092—31no—
66#1949—31no—
67#1948—31no—
68#1875—31no—
69#1874—31no—
70#1854—31no—
71#1853—31no—
72#1804—31no—
73#1802—31no—
74#1798—31no—
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C23×Dic3—21no—
80C3×#121—22no—
81C3×#121Q814yes—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#217—21no—
103#926—41no—
104#926—41no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—41no—
111C2×C4⋊Dic3—41no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124#832—41no—
125#832—41no—
126#832—41no—
127#832—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—22no—
133#830—22no—
134#808—41no—
135#808—41no—
136#808—41no—
137#807—41no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×#121C2×D311yescharacteristic
141C2×C4×D4C2×D311yescharacteristic
142C2×#117—31no—
143C2×#117C2×D311yescharacteristic
144C22×C4⋊C4—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×C4⋊C8—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157#390—21no—
158#390C2×D311yescharacteristic
159#389—61no—
160#389—61no—
161#389C2×D311yescharacteristic
162#388—21no—
163#386—21no—
164#385—21no—
165#385—61no—
166#385—61no—
167#384—31no—
168#384—31no—
169#382—31no—
170#382—31no—
171#380C2×D311yescharacteristic
172C4×C4⋊C4C2×D311yescharacteristic
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183C24×C3—21no—
184C24×C3C2×D411yescharacteristic
185C2×C3×D4—24no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—81no—
192C2×C3×D4—81no—
193C2×C3×D4C2×Q812yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—42no—
202C22×C3×C4—81no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×Q812yes—
210C22×C3×C4#4112yes—
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—81no—
218C22×Dic3—81no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4C2×Q812yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—81no—
233C3×C22⋊C4—81no—
234C3×C22⋊C4—81no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—81no—
238C3×C42#4112yes—
239C3×C42#4112yes—
240#217—42no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251C4⋊Dic3—81no—
252C4⋊Dic3—81no—
253C4⋊Dic3—81no—
254C4×Dic3—42no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C22×D4—21no—
268C22×D4C22×D311yescharacteristic
269C23×C4—21no—
270C23×C4—61no—
271C23×C4C22×D311yescharacteristic
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276#121—22no—
277#121Dic614yes—
278C4×D4—22no—
279C4×D4—22no—
280#117—22no—
281#117—22no—
282#117—121no—
283#117—121no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—121no—
295C2×C4⋊C4—121no—
296C2×C4⋊C4#6711yescharacteristic
297C2×C4⋊C4#6711yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4#6711yescharacteristic
308C2×C22⋊C4#6711yescharacteristic
309C2×C42—21no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—121no—
314#106—121no—
315#106—121no—
316#106—121no—
317C4⋊C8—62no—
318C4⋊C8—62no—
319#102—121no—
320#102—121no—
321#102—121no—
322#102—121no—
323C22⋊C8—62no—
324C22⋊C8—62no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—22no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3C22⋊Q812yes—
346C23×C3C22×D411yescharacteristic
347C3×D4—22no—
348C3×D4—22no—
349C3×D4—41no—
350C3×D4—41no—
351C3×D4—84no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—84no—
378C2×C3×C4C22⋊Q812yes—
379C2×C3×C4#12312yes—
380C2×C3×C4#12312yes—
381C2×C3×C4C22×Q811yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—84no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C24—21no—
405C24C2×#6711yescharacteristic
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4C2×Dic612yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4—241no—
432C22×C4C2×Dic612yes—
433C22×C4#23512yes—
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D311yescharacteristic
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—81no—
468C4⋊C4—122no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4—241no—
473C4⋊C4C2×Dic612yes—
474C22⋊C4—24no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—122no—
482C42—21no—
483C42—21no—
484C42—81no—
485C42—122no—
486C42#23712yes—
487C42#23712yes—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3C2×C22⋊Q811yescharacteristic
503C22×C3C2×#12311yescharacteristic
504C22×C3#53511yescharacteristic
505C22×C3#57611yescharacteristic
506C22×C3#57711yescharacteristic
507C22×C3#57711yescharacteristic
508C22×C3#57711yescharacteristic
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—42no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4#55311yescharacteristic
518C3×C4#55311yescharacteristic
519C3×C4#55411yescharacteristic
520C3×C4#55411yescharacteristic
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#92412yes—
539C23C22×#6711yescharacteristic
540D4—22no—
541D4—22no—
542D4—41no—
543D4—41no—
544D4—84no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—244no—
584C2×C4#92412yes—
585C2×C4#93512yes—
586C2×C4#93512yes—
587C2×C4C22×Dic611yescharacteristic
588C2×C4C2×#23511yescharacteristic
589C2×C4C2×#23711yescharacteristic
590C2×C4C2×#23711yescharacteristic
591C2×C4#100911yescharacteristic
592C2×C4#100911yescharacteristic
593C8—121no—
594C8—121no—
595C8—121no—
596C8—121no—
597C2×C3—42no—
598C2×C3—82no—
599C2×C3#248711yescharacteristic
600C2×C3#308111yescharacteristic
601C2×C3#308411yescharacteristic
602C2×C3#309611yescharacteristic
603C2×C3#309911yescharacteristic
604C2×C3#322111yescharacteristic
605C2×C3#322411yescharacteristic
606C22—22no—
607C22—22no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C2×#92411yescharacteristic
621C22C2×#93511yescharacteristic
622C22#583011yescharacteristic
623C22#583511yescharacteristic
624C22#583611yescharacteristic
625C22#583611yescharacteristic
626C22#583711yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—241no—
636C4—241no—
637C4—241no—
638C4—241no—
639C4#554311yescharacteristic
640C4#554311yescharacteristic
641C4#556511yescharacteristic
642C4#556511yescharacteristic
643C3#2768811yescharacteristic, complemented, Sylow
644C2—42no—
645C2—82no—
646C2#8237411yescharacteristic
647C2#8253911yescharacteristic
648C2#8256011yescharacteristic
649C2#8337611yescharacteristic
650C2#8341911yescharacteristic
651C2#8344011yescharacteristic
652C2#8346311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2324C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2324D3
hypocenterC3#27688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #82233C2semidirect
5. #71605C2semidirect
6. #71604C2semidirect
7. #71583C2semidirect
8. #69656C2semidirect
9. #69654C2semidirect
10. #69613C2semidirect
11. #69611C2semidirect
12. #69506C2semidirect
13. #69505C2semidirect
14. #69468C2semidirect
15. #69467C2semidirect
643. C3#27688semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#23242
#822332
#716782
#716772
#716052
#716042
#715832
#696562
#696542
#696132
#696112
#695062
#695052
#694682
#694672
#276883